Private schools, however, can be selective about their admissions in a way that public schools cannot, taking students that are better prepared to begin with, or have a better support system outside of school to augment their in-school learning. It's far from clear that they achieve better results with lower costs once you control for that factor.
The parent poster's point, I believe, was that public schools, and public roads, are a socialized function in order to achieve universality. There are economic and social benefits to an educated population, and to improved flow of labor and commerce, which accrue to everyone to some degree.
Whether that same universality is a desirable thing for health care is currently a matter of debate. Personally, I think it is, but I don't have it in me to argue the point at this time of night.